Solve the exponential equation for $x$. 2 8 − x 32 x + 12 = 2 3 x − 7 \dfrac{2\^{8-x}}{32\^{ x+12}}=2\^{ 3x-7} $x=$
Solution: The strategy Let's write $32$ in base $2$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $2$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 2 8 − x 32 x + 12 = 2 8 − x ( 2 5 ) x + 12 = 2 8 − x 2 5 x + 60 = 2 8 − x − ( 5 x + 60 ) = 2 − 6 x − 52 ( 32 = 2 5 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned} \dfrac{2\^{8-x}}{32\^{ x+12}}&=\dfrac{2\^{8-x}}{(2^5)\^{ x+12}}&&&&(32=2^5) \\\\\\\\ &=\dfrac{2\^{ C{8-x}}}{2\^{ {5x+60}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=2\^{ C{8-x} \ - \ ({5x+60})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=2\^{ -6x-52} \end{aligned} Solving the equation We obtain the following equation. 2 − 6 x − 52 = 2 3 x − 7 2\^{ -6x-52}=2\^{ 3x-7} Now we can equate the exponents and solve for $x$. $\begin{aligned} -6x-52 &=3x-7\\\\ x &= -5\end{aligned}$ The answer The answer is $x=-5$. You can check this answer by substituting $\it{x=-5}$ in the original equation and evaluating both sides.